Welcome to Telma's Library - Plotting

This Jupyter notebook aims at showing some basic plotting function of Telma.

Plotting functions are located in OneLife or MultiplesLives classes.

Besides the "classic static" manner of plotting in life actuarial science. Telma permits plotting charts interactively.

Let's have a look at the following examples.

import pandas as pd  ## Import the data
from telma import OneLife, MultipleLives ## Import OneLife and MultipleLives

Using G20 dataset.

Note: this database refers to the G20 countries and that was downloaded from kaggle. Click the following link to obtain it. Dataset lifeTables

file = r'C:\Users\Josè Valencia\Desktop\Actuarial2\lifeTable.csv'
dataset = pd.read_csv(file, index_col=0) 
dataset

	Australia_Male	Austria_Male	Belgium_Male	Bulgaria_Male	Canada_Male	Czech Republic_Male	Denmark_Male	Estonia_Male	Finland_Male	France_Male	...	Netherlands_Female	Poland_Female	Portugal_Female	Russia_Female	Slovakia_Female	Slovenia_Female	Spain_Female	Sweden_Female	United Kingdom_Female	United States_Female
age
0	100000	100000	100000	100000	100000	100000	100000	100000	100000	100000	...	100000	100000	100000	100000	100000	100000	100000	100000	100000	100000
1	99528	99600	99620	98938	99488	99698	99689	99582	99740	99586	...	99649	99493	99693	99279	99526	99747	99713	99759	99583	99415
2	99495	99572	99570	98887	99453	99674	99663	99521	99723	99554	...	99624	99461	99657	99205	99486	99727	99682	99735	99558	99376
3	99471	99545	99549	98854	99433	99646	99657	99521	99717	99531	...	99609	99443	99639	99160	99444	99717	99669	99718	99538	99351
4	99458	99528	99528	98795	99421	99637	99654	99521	99707	99513	...	99595	99424	99626	99130	99414	99706	99657	99700	99522	99332
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
106	45	20	13	5	48	4	14	15	18	35	...	75	38	100	14	22	66	134	69	137	198
107	23	10	6	3	25	2	7	9	9	18	...	38	19	54	7	11	34	70	34	75	112
108	12	5	3	1	13	1	3	5	4	9	...	18	10	28	3	6	17	35	16	39	61
109	6	2	1	1	6	0	2	2	2	4	...	9	5	14	2	3	8	17	8	20	32
110	3	1	1	0	3	0	1	1	1	2	...	4	2	7	1	1	4	8	3	10	16

111 rows × 56 columns

Usa_male and Usa_females cohorts.

usa_male = dataset['United States_Male']
usa_female = dataset['United States_Female']

---

Plotting using OneLife and MultipleLives (statically).

---

OneLife

It will be possible to graph either from a cohort function or several.

• For plotting one cohort, OneLife will be used.
  
• For plotting more than one cohorts, MultipleLives. will be used.
  

Example with one cohort: Usa_Male.

Invoking OneLife, lifeplot method has as parameters:

• cohort1. It refers to the cohort to be passed.
• interactive. If true an interactive chart will be plotted using Plotly.
• death. It refers to the numbers of death chart $d(x)$, if True when plotting interactively it will show the number of death chart.
• qx_log. It refers to the probability of death in logarithmic scale, that is $log(q(x))$. If True when plotting interactively it will show $log(q(x))$.

usa_male

age
0      100000
1       99284
2       99237
3       99206
4       99182
        ...  
106        51
107        27
108        14
109         7
110         3
Name: United States_Male, Length: 111, dtype: int64

OneLife().lifeplot(usa_male) ## By default. lx and dx charts will be plotted.

Hump mortality of usa_male

OneLife().lifeplot(usa_female, qx_log=True) ## If qx_log=True (and interactively=False), log(q(x)) will be plotted.

MultipleLives

The question: do I want to plot more cohorts?

Follow the same steps as OneLife ().lifeplot but with:

• MultipleLives $\rightarrow$ lifeplot_2 method
• Two cohorts. That is, by default at least two parameters will be needed; $ l_x $ and $l_y$ the cohort functions
• A third cohort can be passed, if perhaps you want to confront a model.

Example: usa_male, usa_female and a "model".

The model will be specified as:

$$F(X) = 1- e^{-0.0001x^{e^{x/115}}}$$$$S(X) = 1-F(X)= e^{-0.0001x^{e^{x/115}}}$$

$Clearly:$

\begin{equation} 1- S(X) = F(X)= \begin{cases} 0, & \text{if}\ x < 0 \\ 1 - e^{-0.0001x^{e^{x/115}}}, & \text{if}\ 0 \le X < 115\\ 1, & \text{if}\ x \geq\ 115 \end{cases} \end{equation} $F(115) = 1$ if $1 - e^{(-0.0001 \cdot 115^{e^{(115/115)}})}.$ Consequently: $S(115) = 0$.

import numpy as np
def vettore():  ## The "ficticious model"
    x = np.arange(0,115)
    numera = np.exp(-.0001*x**(np.exp(x/115)))
    l = pd.Series(numera*10**5)
    vettore = l[0:-8].round()
    return vettore

vettore()

0      100000.0
1       99990.0
2       99980.0
3       99969.0
4       99958.0
         ...   
102        54.0
103        20.0
104         7.0
105         2.0
106         0.0
Length: 107, dtype: float64

## length of the three cohorts 
len(usa_male), len(usa_female), len(vettore())

(111, 111, 107)

## Graph of usa_male, usa_female and the dummy model with different length lx
MultipleLives().lifeplot_2(cohort1=usa_male, cohorte2=usa_female, model=vettore())

NOTE: Dummy model has no mortality hump around 20s

MultipleLives().lifeplot_2(cohort1=usa_male, model=vettore(),
                           cohorte2 = usa_female, qx_log=True) 

---

Plotting using OneLife and MultipleLives (dinamically).

---

Now The question is: do I want to see the graph in a dynamic and interactive way?

I want to see it in interactive mode, for more cohorts, and tell me the life expectancy? Example, with the French:

Passing the mouse on it you can observe onto the graph:

• $ Variable: $ It refers to the desired observed cohort
• $ Age: $ Referred to the age of the observed variable in question.
• $ value: $ Referred to living, that is, to the survivors of the cohort visualized in question.
• $e_x1$, $e_x2$, $e_x3$: These are the life expectancies ("complete"), respectively, of the cohort variables rounded to 2 decimal places.
  The order of the passed parameters is followed, that is, ex3 is the life expectancy of French women, ex2 US women, etc.

Note: From here the charts are interactive. It is requested that you pass the mouse onto it to see in detail.

france = dataset.France_Female
france.head()# vector lx de las mujeres francesas

age
0    100000
1     99664
2     99638
3     99619
4     99607
Name: France_Female, dtype: int64

MultipleLives().lifeplot_2(usa_male, usa_female, france,
                      interactive=True) # The parameter "interactive = True" is required.
        

Mortality hump: usa_male, france_female and the model

MultipleLives().lifeplot_2(usa_male, model=vettore(), cohorte2 = usa_female, interactive=True, death=True, qx_log= True)

## Notice that the parameters death = True and qx_log = True are passed.
## "Value" in the graph is the qx of the observed cohort in question.
## While Death indicates the deaths at the respective ages that occur in all 3 cohorts. That is dx

MultipleLives().lifeplot_2(usa_male, model=vettore(), cohorte2 = usa_female, interactive=True, death=True, qx_log= False)

## This time values are the deaths of the cohorts in question. And the probabilities qx are observed

If the length of $l(x)$ is different from $l(y)$ consider passing the vector with less "len" in model parameter. (It might be not necessary)